Let X and Y be real Banach spaces and $varepsilon$, p $geq$ 0. A mapping T between X and Y is called an ($varepsilon$, p)-isometry if ｜∥T(x)-T(y)∥-∥x-y∥｜$leq$ $varepsilon$∥x-y∥$^{p}$ for x, y$in$X. Let H be a real Hilbert space and T : H longrightarrow H an ($varepsilon$, p)-isometry with T(0) = 0. If p$ eq$1 is a nonnegative number, then there exists a unique isometry I : H longrightarrow H such that ∥T(x)-I(y)∥$leq$ C($varepsilon$)(∥x∥$^{ 1+p)/2}$+∥x∥$^{p}$ ) for all x$in$H, where C($varepsilon$) longrightarrow 0 as $varepsilon$ longrightarrow 0.